The existence results are proven based on the earlier mentioned result for base polyhedra proven in earlier work [1]} and on the relation of the aforementioned sets with these polyhedra. For instance, the existence of least weakly \((a,b)\) -supermajorized elements for submodular polyhedra can be proven using the fact that such polyhedra contain vectors that are in some sense maximal and thus necessarily minimize any non-increasing function over such polyhedra. These maximal vectors form a base polyhedron within the given submodular polyhedron. Furthermore, the existence of least weakly absolutely \((a,b)\) -majorized elements for bisubmodular polyhedra uses the fact that such polyhedra are the intersection of (reflections of) particular submodular polyhedra and contain all corresponding (reflected) base polyhedra (see also [2]}, [3]}).
